Finding Like Radicals A Step By Step Guide To Simplifying Cube Root Of 54

Hey there, math enthusiasts! Ever stumbled upon a radical expression and felt a bit lost in the maze of numbers and roots? Well, you're not alone! Radicals can seem intimidating at first, but with a little practice and the right approach, they become much easier to handle. Today, we're diving into the world of simplifying radicals, specifically focusing on identifying like radicals. We'll take a close look at the expression $\sqrt[3]{54}$ and figure out which of the given options is its like radical after simplification. So, grab your thinking caps, and let's get started!

Understanding Like Radicals

Before we jump into simplifying radicals and tackling our specific problem, let's make sure we're all on the same page about what like radicals actually are. Like radicals are radical expressions that have the same index (the small number indicating the type of root, like the '3' in a cube root) and the same radicand (the number or expression under the radical symbol). Think of it like this: like radicals are like terms in algebra – you can combine them through addition and subtraction, making them super useful in simplifying expressions. For example, $2\sqrt{3}$ and $5\sqrt{3}$ are like radicals because they both have a square root (index of 2, usually not written) and the same radicand, 3. However, $\sqrt{2}$ and $\sqrt{3}$ are not like radicals because they have different radicands, even though they have the same index. Similarly, $\sqrt{5}$ and $\sqrt[3]{5}$ are not like radicals because they have different indices. Understanding this concept is crucial because it guides how we simplify and compare radicals. We're essentially looking for expressions that, after being simplified, match both the index and the radicand of our original expression. This is why simplifying is so important – it allows us to reveal the underlying structure of the radical and see if it matches another radical's simplified form. Without simplifying, it's like trying to compare apples and oranges; with simplifying, we can break down the radicals into their core components and see if they're actually the same fruit, just presented differently. So, keep the concept of like radicals in mind as we move forward, because it's the key to unlocking this type of problem!

Simplifying $\sqrt[3]{54}$

Okay, guys, let's get down to business and simplify $\sqrt[3]{54}$! The key to simplifying radicals lies in finding the largest perfect cube (since we're dealing with a cube root) that divides evenly into the radicand, which is 54 in this case. A perfect cube is a number that can be obtained by cubing an integer (e.g., 8 is a perfect cube because $2^3 = 8$). So, we need to think about the perfect cubes: 1, 8, 27, 64, and so on. Which of these divides evenly into 54? You got it – it's 27! Now, we can rewrite 54 as the product of 27 and 2, so we have $\sqrt[3]{54} = \sqrt[3]{27 \cdot 2}$. The beauty of this is that we can now use the property of radicals that says the cube root of a product is the product of the cube roots: $\sqrt[3]{27 \cdot 2} = \sqrt[3]{27} \cdot \sqrt[3]{2}$. And what's the cube root of 27? It's 3, of course! So, we have $\sqrt[3]{27} \cdot \sqrt[3]{2} = 3\sqrt[3]{2}$. There you have it! We've successfully simplified $\sqrt[3]{54}$ to $3\sqrt[3]{2}$. This is our target form – we're looking for other radicals that, when simplified, will also have a radicand of 2 and an index of 3. This simplified form makes it much easier to compare with the other options and identify the like radical. Remember, the goal is to break down the radical into its simplest form, where the radicand has no more perfect cube factors. This process not only helps us identify like radicals but also makes further calculations and operations with radicals much smoother. So, let's keep this simplified form in mind as we evaluate the other options and see which one matches our $3\sqrt[3]{2}$.

Evaluating the Options

Now that we've simplified $\sqrt[3]{54}$ to $3\sqrt[3]{2}$, let's dive into the options and see which one is a like radical. We'll go through each option one by one, applying the same simplification techniques we used before. Remember, we're looking for a radical that, when simplified, will have the same index (3) and the same radicand (2) as our simplified expression.

Option 1: $\sqrt[3]{24}$

Let's tackle $\sqrt[3]{24}$ first. We need to find the largest perfect cube that divides evenly into 24. Think about the perfect cubes again: 1, 8, 27... Aha! 8 is a factor of 24. We can rewrite 24 as $8 \cdot 3$. So, $\sqrt[3]{24} = \sqrt[3]{8 \cdot 3}$. Using the property of radicals, we get $\sqrt[3]{8 \cdot 3} = \sqrt[3]{8} \cdot \sqrt[3]{3}$. And the cube root of 8 is 2, so we have $2\sqrt[3]{3}$. Notice anything? This simplifies to $2\sqrt[3]{3}$, which has a radicand of 3, not 2. So, $\sqrt[3]{24}$ is not a like radical to $\sqrt[3]{54}$.

Option 2: $\sqrt[3]{162}$

Next up is $\sqrt[3]{162}$. This one looks a bit larger, but the process is the same. We need to find the largest perfect cube that divides 162. Let's run through the perfect cubes: 1, 8, 27, 64... 27 seems promising. Let's see if 27 divides 162. And yes, it does! 162 divided by 27 is 6. So, we can rewrite 162 as $27 \cdot 6$. Thus, $\sqrt[3]{162} = \sqrt[3]{27 \cdot 6}$. Applying the property of radicals, we get $\sqrt[3]{27 \cdot 6} = \sqrt[3]{27} \cdot \sqrt[3]{6}$. The cube root of 27 is 3, so we have $3\sqrt[3]{6}$. Hmm, the radicand here is 6, not 2. So, $\sqrt[3]{162}$ is also not a like radical to $\sqrt[3]{54}$. But don't worry, we're learning with each step!

Option 3: $\sqrt{128}$

Now we have $\sqrt{128}$. Notice something different about this one? It's a square root, not a cube root! This means the index is 2, not 3. Even if we simplify it, it can't be a like radical to $\sqrt[3]{54}$ because they have different indices. Just to illustrate, let's simplify it anyway. We need to find the largest perfect square that divides 128. The perfect squares are 1, 4, 9, 16, 25, 36, 49, 64... 64 works! 128 can be rewritten as $64 \cdot 2$. So, $\sqrt{128} = \sqrt{64 \cdot 2} = \sqrt{64} \cdot \sqrt{2} = 8\sqrt{2}$. As expected, it's a square root with a radicand of 2, but it's not a cube root. So, this is definitely not a like radical.

Option 4: $\sqrt[3]{128}$

Finally, we have $\sqrt[3]{128}$. This is a cube root, so it's in the running! Let's find the largest perfect cube that divides 128. Going through the list: 1, 8, 27, 64... 64 is a perfect cube ($4^3 = 64$) and it divides 128! 128 divided by 64 is 2. So, we can rewrite 128 as $64 \cdot 2$. Thus, $\sqrt[3]{128} = \sqrt[3]{64 \cdot 2} = \sqrt[3]{64} \cdot \sqrt[3]{2}$. The cube root of 64 is 4, so we have $4\sqrt[3]{2}$. Bingo! This simplifies to $4\sqrt[3]{2}$, which has the same index (3) and the same radicand (2) as our simplified expression for $\sqrt[3]{54}$, which was $3\sqrt[3]{2}$.

The Verdict: Identifying the Like Radical

Alright, we've done the detective work, and the results are in! After carefully simplifying each option, we found that $\sqrt[3]{128}$ is the like radical to $\sqrt[3]{54}$. Remember, the key was to simplify each radical expression and then compare their simplified forms. We broke down each radicand into its prime factors, identified the largest perfect cube factors, and pulled them out of the radical. This allowed us to see the underlying structure of each radical and easily identify the one that matched our target. It's like unwrapping a gift to see what's inside – simplifying radicals reveals their true nature and makes comparisons straightforward. So, the final answer is $\sqrt[3]{128}$, which simplifies to $4\sqrt[3]{2}$, making it a like radical to $3\sqrt[3]{2}$, the simplified form of $\sqrt[3]{54}$. Great job, guys! You've successfully navigated the world of simplifying radicals and identifying like radicals. Keep practicing, and you'll become a radical-simplifying pro in no time!